CIRM - Videos & books Library - Definable convex and henselian valuations on ordered fields

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Authors : Krapp, Lothar Sebastian (Author of the conference)
CIRM (Publisher )

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Abstract : A valuation $v$ on a field $K$ is said to be definable (in a specified language) if its corresponding valuation ring is a definable subset of $K$. Historically, the study of definable valuations on certain fields was motivated by the general analysis of definable subsets of fields and related decidability questions, but has also re-emerged lately in the context of classifying NIP fields. In my talk, I will present some recent progress in the study of definable valuations on ordered fields ([1] to [4]), where definability is considered in the language of rings as well as the richer language of ordered rings. Within this framework, the focus lies on convex valuations, that is, valuations whose valuation ring is convex with respect to the linear ordering on the field. The most important examples of such valuations are the henselian ones, which are convex with respect to any linear ordering on the field. I will present topological conditions on the value group and the residue field ensuring the definability of the corresponding valuation. Moreover, I will outline some definability and non-definability results in the context of specific classes of ordered fields such as t-henselian, almost real closed, and strongly dependent ones.

Keywords : definable valuations; ordered fields; convex valuations; henselian valuations; almost real closed fields

MSC Codes :
03C64 - Model theory of ordered structures; o-minimality
12J10 - Valued fields
12J25 - Non-Archimedean valued fields, See also {30G06, 32P05, 46S10, 47S10}
13F25 - Formal power series rings, See also {13J05}
13J15 - Henselian rings, See also {13B40}
13J30 - Real algebra

Additional resources :
https://webusers.imj-prg.fr/~zoe.chatzidakis/CIRM/Krapp.pdf

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2023-06-01_Krapp.mp4

Information on the Event

Event Title : Model theory of valued fields / Théorie des modèles des corps valués
Event Organizers : Chatzidakis, Zoé ; Jahnke, Franziska ; Rideau-Kikuchi, Silvain
Dates : 29/05/2023 - 02/06/2023
Event Year : 2023
Event URL : https://conferences.cirm-math.fr/2761.html

Citation Data

DOI : 10.24350/CIRM.V.20052503
Cite this video as: Krapp, Lothar Sebastian (2023). Definable convex and henselian valuations on ordered fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20052503
URI : http://dx.doi.org/10.24350/CIRM.V.20052503

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Bibliography

DITTMANN, Philip, JAHNKE, Franziska, KRAPP, Lothar Sebastian, et al. Definable valuations on ordered fields. arXiv preprint arXiv:2206.15301, 2022. - https://doi.org/10.48550/arXiv.2206.15301

KRAPP, Lothar Sebastian, KUHLMANN, Salma, et LEHÉRICY, Gabriel. Ordered fields dense in their real closure and definable convex valuations. In : Forum Mathematicum. De Gruyter, 2021. p. 953-972. - https://doi.org/10.1515/forum-2020-0030

KRAPP, Lothar Sebastian, KUHLMANN, Salma, et LEHÉRICY, Gabriel. Strongly NIP almost real closed fields. Mathematical Logic Quarterly, 2021, vol. 67, no 3, p. 321-328. - https://doi.org/10.1002/malq.202000060

KRAPP, Lothar Sebastian, KUHLMANN, Salma, et LINK, Moritz. Definability of henselian valuations by conditions on the value group. The Journal of Symbolic Logic, 2022, p. 1-19. - https://doi.org/10.1017/jsl.2022.34

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